Knot groups, quandle extensions and orderability

TL;DR

This paper explores the relationship between knot groups, quandle extensions, and orderability, providing new characterizations of 3-manifolds and analyzing the orderability properties of link quandles and their extensions.

Contribution

It introduces a novel characterization of L-space 3-manifolds using quandle orderability and investigates orderability conditions of dynamical quandle extensions.

Findings

Characterization of L-space 3-manifolds via quandle orderability

Conditions for conjugation quandle orderability in group extensions

Non-orderability of certain link quandles and existence of orderable enveloping groups

Abstract

This paper gives a new way of characterizing L-space $3$-manifolds by using orderability of quandles. Hence, this answers a question of Adam Clay et al. [Question 1.1 of Canad. Math. Bull. 59 (2016), no. 3, 472-482]. We also investigate both the orderability and circular orderability of dynamical extensions of orderable quandles. We give conditions under which the conjugation quandle on a group, as an extension of the conjugation of a bi-orderable group by the conjugation of a right orderable group, is right orderable. We also study the right circular orderability of link quandles. We prove that the $n$-quandle $Q_n(L)$ of the link quandle of $L$ is not right circularly orderable and hence it is not right orderable. But on the other hand, we show that there are infinitely many links for which the $p$-enveloping group of the link quandle is right circularly orderable for any prime integer $p$.